Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 10:10:11 -0500

robert j. kolker said:
>
>
> Tony Orlow (aeo6) wrote:
>
> >
> >
> > Albert -
> >
> > The whole concept of Cantor's cardinality of infinite sets lies in
> > drawing 1-1 correspondences between members of the sets.
>
> The 1-1 correspondences are between the -sets- not members of the sets.
> One set is the domain of a 1-1 onto function, the other is the co-domain.

Bob stop being a jerk. First off, the 1-1 mapping is between the sets of
members, one member in the first set mapped to exactly one member in the
other set and vice versa. Talk about semantic blabbage! There nothing
wrong with what I said.

>
>
> > He determined that sets like the integers or rationals have the same cardinality by
> > this definition,
>
> This was -proven= by producting an explicit function which mapped the
> set of integers one to one onto the set of rationals. Its a theorem, not
> a definition
The definition of cardinality for infinite sets is based on the 1-1
correspondence between set members. It draws a distinction between
levels of infinity, but not a complete distinction between all
infinities. If cardinality of such sets is vague, perhaps we need to
define a different measure that handles differences between infinities
that are on scales of less than infinity^infinity.
>
>
>
> > but that reals have a higher cardinality because there
> > is no way to form a mapping between the reals and integers. In other
> > words, the infinities that enumerate the integers or rationals are a
> > smaller type of infinity than the infinity that enumerates the reals. I
> > have no problem with this. there are definitely more reals than there
> > are integers or rationals. After all, there are an infinite number of
> > reals between any two integers.
> >
> > My problem lies in the interpretation that cardinality in this sense is
> > an exact size of the set. For finite sets, cardinality is exactly the
> > number of members in the set. When dealing with infinite sets, it's
> > impossible to say "exactly" infinity, since there are a wide range of
> > infinities. Cardinality as Cantor proposed is a good start, but I
> > disagree with statements like "there are equally many even integers as
> > inetegers in general." That's where I disagree. there are half as many
> > evens as integers.
>
> The set of even integers can be put into 1 - 1 correspondence with the
> integers by the function n <-> 2*n.
>
> You are wrong. You seem to have trouble with the idea of a one to one
> correondence, but if you have twenty pennies in your pocket you would
> have no trouble matching the pennies up with pebbles from a set of 20
> pebbles. Why is that? Children can do one to one match-ups. Why can't you?
>
> Bob Kolker
>
I don't have a problem with 1-1 correspondence. You have a problem
listening to anything that didn't originate from your own head. 

-- 
Smiles,
Tony

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